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Claim. Let $f(z)=\sum_{k\ge1}a_kz^{n_k}$ be an entire function with $a_k\ne0$ for every $k$. Miécislas Biernacki proves that if $\sum_k1/n_k<\infty$, then $f$ takes every complex value infinitely often. The site cites the result as [Bi28], Sur les équations algébriques contenant des paramètres arbitraires (1928), 145 pages, which Erdős [Er61] cites as Biernacki's Paris thesis of 1928; the thesis text appeared in Bull. Int. Acad. Polon. Sci. Lett. Sér. A (1927), 542--685, and a note, Sur les fonctions entières à séries lacunaires, in C. R. Acad. Sci. Paris 187 (1928), 477--479, both cited in Murai's bibliography. This page is dated by the earliest of these records. The statement follows the site's account and the account in Murai's paper (library card), which calls it the Fejér--Biernacki theorem; neither publication is held in this repository, and the proof is not reconstructed here. Fejér [Fe08] had proved under the same hypothesis that every value is taken at least once. The formal-conjectures statement file of the problem states this theorem as erdos_517.variants.fejer, tagged research solved and credited to [Bi28], with its proof left open (pinned); a statement file is not a formalization of the result.

Covers. Every instance of Problem 517 with $\sum_k1/n_k<\infty$, of any order. Such exponents satisfy the question's hypothesis: by the Cauchy criterion $\sum_{k/2<j\le k}1/n_j\ge(k/2)/n_k$ tends to $0$, so $n_k/k\to\infty$. The functions with $n_k/k\to\infty$ but $\sum1/n_k=\infty$ are not covered; for those of finite order Pólya 1929 gives the answer, and those of infinite order remain open.

Depends on. Nothing in this wiki: the theorem is Biernacki's own.

Standing. Pending. The theorem is classical and Murai 1983 proves a stronger statement in a refereed journal, but no evidence that the Bulletin of the Polish Academy or the Comptes Rendus refereed Biernacki's publications is recorded, so refereed is not listed. The site credits the theorem in its commentary on a problem it labels OPEN, which is not acceptance of a solution, so reviewed is not listed either.